| L(s) = 1 | + (1.68 − 0.386i)3-s + (−0.477 − 2.70i)5-s + (−1.82 − 1.52i)7-s + (2.70 − 1.30i)9-s + (0.0434 − 0.246i)11-s + (−2.45 − 0.893i)13-s + (−1.85 − 4.38i)15-s + (0.146 + 0.254i)17-s + (−1.39 + 2.41i)19-s + (−3.66 − 1.87i)21-s + (5.12 − 4.30i)23-s + (−2.40 + 0.876i)25-s + (4.05 − 3.24i)27-s + (0.333 − 0.121i)29-s + (−2.11 + 1.77i)31-s + ⋯ |
| L(s) = 1 | + (0.974 − 0.223i)3-s + (−0.213 − 1.21i)5-s + (−0.688 − 0.577i)7-s + (0.900 − 0.434i)9-s + (0.0130 − 0.0742i)11-s + (−0.680 − 0.247i)13-s + (−0.478 − 1.13i)15-s + (0.0355 + 0.0616i)17-s + (−0.319 + 0.553i)19-s + (−0.799 − 0.409i)21-s + (1.06 − 0.896i)23-s + (−0.481 + 0.175i)25-s + (0.780 − 0.624i)27-s + (0.0619 − 0.0225i)29-s + (−0.380 + 0.319i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0929 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0929 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22632 - 1.11722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22632 - 1.11722i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.68 + 0.386i)T \) |
| good | 5 | \( 1 + (0.477 + 2.70i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (1.82 + 1.52i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.0434 + 0.246i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (2.45 + 0.893i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.146 - 0.254i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.39 - 2.41i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.12 + 4.30i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.333 + 0.121i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.11 - 1.77i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.49 - 6.05i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.13 - 3.32i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0452 - 0.256i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.75 - 7.34i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 5.43T + 53T^{2} \) |
| 59 | \( 1 + (1.03 + 5.88i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (9.07 + 7.61i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.70 - 0.619i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (0.185 + 0.320i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.51 - 4.35i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.754 + 0.274i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (2.58 - 0.942i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (5.22 - 9.05i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.57 - 14.6i)T + (-91.1 - 33.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77807588623130710722815587883, −9.782498745966529680518753397830, −9.092040089108280056904929976139, −8.257116202118847933765838298788, −7.44501286534387694714165424462, −6.39720463165659673118012149804, −4.88550648127576081200875966015, −3.98526776792184608331104546896, −2.73154452989583862208565013108, −1.00243087488254085998436137029,
2.39963629794728371896681502478, 3.09926662392709452389310410958, 4.24940257157318450393275860838, 5.75174837913568860590348321719, 7.08952799294694781762300347006, 7.42002391976118509002172913703, 8.893306387664340801675005182368, 9.425603780743466538186358184455, 10.40722159017584741809851276720, 11.17406607798010684420739968017
